Lerch zeta function

In mathematics, the Lerch zeta-function, sometimes called the Hurwitz-Lerch zeta-function, is a special function that generalizes the Hurwitz zeta-function and the polylogarithm. It is named after Mathias Lerch [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lerch.html] .

Definition

The Lerch zeta-function is given by

:$L(lambda,\; alpha,\; s)\; =\; sum\_\{n=0\}^inftyfrac\; \{\; exp\; (2pi\; ilambda\; n)\}\; \{(n+alpha)^s\}.$

A related function, the Lerch transcendent, is given by

:$Phi(z,\; s,\; alpha)\; =\; sum\_\{n=0\}^inftyfrac\; \{\; z^n\}\; \{(n+alpha)^s\}.$

The two are related, as

:$,Phi(exp\; (2pi\; ilambda),\; s,alpha)=L(lambda,\; alpha,s).$

Integral representations

An integral representation is given by

:$Phi(z,s,a)=frac\{1\}\{Gamma(s)\}int\_\{0\}^\{infty\}frac\{t^\{s-1\}e^\{-at\{1-ze^\{-t,dt$

for

:

A contour integral representation is given by

:$Phi(z,s,a)=-frac\{Gamma(1-s)\}\{2pi\; i\}int\_\{0\}^\{(+infty)\}frac\{(-t)^\{s-1\}e^\{-at\{1-ze^\{-t,dt$

for

:

where the contour must not enclose any of the points $t=log(z)+2kpi\; i,kin\; Z.$

A Hermite-like integral representation is given by

:$Phi(z,s,a)=frac\{1\}\{2a^s\}+int\_\{0\}^\{infty\}frac\{z^\{t\{(a+t)^\{s,dt+frac\{2\}\{a^\{s-1int\_\{0\}^\{infty\}frac\{sin(sarctan(t)-talog(z))\}\{(1+t^2)^\{s/2\}(e^\{2pi\; at\}-1)\},dt$

for

:

and

:$Phi(z,s,a)=frac\{1\}\{2a^s\}+frac\{log^\{s-1\}(1/z)\}\{z^a\}Gamma(1-s,alog(1/z))+frac\{2\}\{a^\{s-1int\_\{0\}^\{infty\}frac\{sin(sarctan(t)-talog(z))\}\{(1+t^2)^\{s/2\}(e^\{2pi\; at\}-1)\},dt$

for :$Re(a)>0.$

pecial cases

The Hurwitz zeta-function is a special case, given by:$,zeta(s,alpha)=L(0,\; alpha,s)=Phi(1,s,alpha).$

The polylogarithm is a special case of the Lerch Zeta, given by :$,\; extrm\{Li\}\_s(z)=zPhi(z,s,1).$

The Legendre chi function is a special case, given by:$,chi\_n(z)=2^\{-n\}z\; Phi\; (z^2,n,1/2).$

The Riemann zeta-function is given by:$,zeta(s)=Phi\; (1,s,1).$

The Dirichlet eta-function is given by:$,eta(s)=Phi\; (-1,s,1).$

Identities

For λ rational, the summand is a root of unity, and thus $L(lambda,\; alpha,\; s)$ may be expressed as a finite sum over the Hurwitz zeta-function.

Various identities include::$Phi(z,s,a)=z^n\; Phi(z,s,a+n)\; +\; sum\_\{k=0\}^\{n-1\}\; frac\; \{z^k\}\{(k+a)^s\}$

and

:$Phi(z,s-1,a)=left(a+zfrac\{partial\}\{partial\; z\}\; ight)\; Phi(z,s,a)$

and

:$Phi(z,s+1,a)=-,frac\{1\}\{s\}frac\{partial\}\{partial\; a\}\; Phi(z,s,a).$

eries representations

A series representation for the Lerch transcendent is given by

:$Phi(z,s,q)=frac\{1\}\{1-z\}\; sum\_\{n=0\}^infty\; left(frac\{-z\}\{1-z\}\; ight)^nsum\_\{k=0\}^n\; (-1)^k\; \{n\; choose\; k\}\; (q+k)^\{-s\}.$

The series is valid for all "s", and for complex "z" with Re("z")<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.

A Taylor's series in the first parameter was given by Erdélyi. It may be written as the following series, which is valid for ::$Phi(z,s,a)=z^\{-a\}left\; [Gamma(1-s)left(-log\; (z)\; ight)^\{s-1\}+sum\_\{k=0\}^\{infty\}zeta(s-k,a)frac\{log^\{k\}(z)\}\{k!\}\; ight]$

:"(the correctness of this formula is disputed, please see the )"Please see:B. R. Johnson,Generalized Lerch zeta-function.Pacific J. Math. 53, no. 1 (1974), 189–193."http://projecteuclid.org/Dienst/UI/1.0/Display/euclid.pjm/1102911791?abstract= "

If s is a positive integer, then:$Phi(z,n,a)=z^\{-a\}left\{sum\_$k=0}atop k eq n-1}^{infty}zeta(n-k,a)frac{log^{k}(z)}{k!}+left [Psi(n)-Psi(a)-log(-log(z)) ight] frac{log^{n-1}(z)}{(n-1)!} ight}.

A Taylor series in the third variable is given by:

Series at "a" = -"n" is given by:$Phi(z,s,a)=sum\_\{k=0\}^\{n\}frac\{z^k\}\{(a+k)^s\}+z^nsum\_\{m=0\}^\{infty\}(1-m-s)\_\{m\}Li\_\{s+m\}(z)frac\{(a+n)^m\}\{m!\};\; a\; ightarrow-n$A special case for "n" = 0 has the following series:$Phi(z,s,a)=frac\{1\}\{a^s\}+sum\_\{m=0\}^\{infty\}(1-m-s)\_\{m\}Li\_\{s+m\}(z)frac\{a^\{m${m!}; |a|<1.

An asymptotic series for $s\; ightarrow-infty$:$Phi(z,s,a)=z^\{-a\}Gamma(1-s)sum\_\{k=-infty\}^\{infty\}\; [2kpi\; i-log(z)]\; ^\{s-1\}e^\{2kpi\; ai\}$for and:$Phi(-z,s,a)=z^\{-a\}Gamma(1-s)sum\_\{k=-infty\}^\{infty\}\; [(2k+1)pi\; i-log(z)]\; ^\{s-1\}e^\{(2k+1)pi\; ai\}$for

An asymptotic series in the incomplete Gamma function:$Phi(z,s,a)=frac\{1\}\{2a^s\}+frac\{1\}\{z^a\}sum\_\{k=1\}^\{infty\}frac\{e^\{-2pi\; i(k-1)a\}Gamma(1-s,a(-2pi\; i(k-1)-log(z)))\}\; \{(-2pi\; i(k-1)-log(z))^\{1-s+frac\{e^\{2pi\; ika\}Gamma(1-s,a(2pi\; ik-log(z)))\}\{(2pi\; ik-log(z))^\{1-s$for

References

* Mathias Lerch, "Démonstration élémentaire de la formule: $frac\{pi^2\}\{sin^2\{pi\; x=sum\_\{\; u=-infty\}^\{infty\}frac\{1\}\{(x+\; u)^2\}$", (1903), L'Enseignement Mathématique, 5, pp.450-453.
* M. Jackson, "On Lerch's transcendent and the basic bilateral hypergeometric series $,\_2psi\_2$", (1950) J. London Math. Soc., 25, pp. 189-196
* H. Bateman, "Higher Transcendental Functions", (1953) McGraw-Hill, New York.
* A. Erdélyi, "Higher Transcendental Functions", (1953) McGraw-Hill, New York.
* Ramunas Garunkstis, " [http://www.mif.vu.lt/~garunkstis Home Page] " (2005) "(Provides numerous references and preprints.)"
* Jesus Guillera and Jonathan Sondow, " [http://arxiv.org/abs/math.NT/0506319 Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent] " (2005) "(Includes various basic identities in the introduction.)"
* Ramunas Garunkstis, " [http://www.mif.vu.lt/~garunkstis/preprintai/approx.pdf Approximation of the Lerch Zeta Function] " (PDF)
* Sergej V. Aksenov and Ulrich D. Jentschura, " [http://aksenov.freeshell.org/lerchphi.html C and Mathematica Programs for Calculation of Lerch's Transcendent] " (2002)

* S. Kanemitsu, Y. Tanigawa and H. Tsukada, " [http://www.iisc.ernet.in/nias/HRJ/vol27/Ktt.pdf A generalization of Bochner's formula] ", (undated, 2005 or earlier)

* A. Laurinv cikas,R. Garunkv stis, "The Lerch zeta-function.", Kluwer Academic Publishers, Dordrecht, 2002. viii+189 pp.